CONSISTENCY OF PENALIZED MLE FOR NORMAL MIXTURES IN MEAN AND VARIANCE Running Title: Consistency of Estimates in Normal Mixture By

The finite mixture of normal distributions in both mean and variance parameters have wide applications. It is well known that the likelihood function of this model is unbounded for any given sample size. Hence, the ordinary maximum likelihood estimator is not consistent. At the same time, a local maximum of the likelihood function can often be found to have good statistical properties. In this paper, by introducing a simple penalty function on the component variance parameters, we prove that the penalized maximum likelihood estimator is asymptotically consistent and efficient. The finite sample property of the new estimator is demonstrated through simulations. A genetic data example is also included.